$S$-wave resonance contributions to the $B^0_{(s)}\to \eta_c{(2S)}\pi^+\pi^-$ in the perturbative QCD factorization approach

By employing the perturbative QCD (PQCD) factorization approach, we study the quasi-two-body $B^0_{(s)}\to \eta_c{(2S)}\pi^+\pi^-$ decays, where the pion pair comes from the $S$-wave resonance $f_0(X)$. The Breit$-$Wigner formula for the $f_0(500)$ and $f_0(1500)$ resonances, and the Flatt\'e model for the $f_0(980)$ resonance are adopted to parameterize the time-like scalar form factors in the two-pion distribution amplitudes. As a comparison, Bugg's model is also used for the wide $f_0(500)$ in this work. For decay rates, we found the following PQCD predictions: (a) $ {\cal B}(B^0_s\to \eta_c(2S) f_0(X)[\pi^+\pi^-]_s )=\left ( 2.67^{+1.78}_{-1.08} \right )\times 10^{-5}$ when the contributions from $f_0(980)$ and $f_0(1500)$ are all taken into account; (b) ${\cal B}(B^0\to \eta_c(2S) f_0(500)[\pi^+\pi^-]_s)= \left ( 1.40 ^{+0.92}_{-0.56} \right ) \times 10^{-6}$ in the Breit-Wigner model and $ \left ( 1.53 ^{+0.97}_{-0.61} \right ) \times 10^{-6}$ in the Bugg's model.

The PQCD factorization approach is one of the major theoretical frameworks to deal with the two-body hadronic B meson decays [61,62]. Very recently, some three-body hadronic B meson decays have been studied by employing the PQCD factorization approach, for example in Refs. [39][40][41][42][43][44][45][46][47]. For the cases of three-body decays, however, the previous PQCD approach [61,62] should be modified by introducing the two-meson distribution amplitudes [63][64][65][66] to describe the selected pair of final state mesons due to the following reason discussed in [61,62]: the contribution from the direct evaluation of hard b-quark decay kernels containing two virtual gluons is generally power suppressed, and the dominant contribution comes most possibly from the region where the two energetic light mesons are almost collimating to each other with an invariant mass below O(Λm B )(Λ = m B − m b , means the B meson and b quark mass difference). Then, the typical PQCD factorization formula with the crucial nonperturbative input of two-hadron distribution amplitudes for a B → h 1 h 2 h 3 decay amplitude can be written symbolically in the form of Here the hard kernel H(x i , b i , t) contains the contributions from one hard gluon exchange diagrams only, the nonper- are the distribution amplitudes for the B meson, the h 1 -h 2 pair and the h 3 meson respectively, while the symbols ⊗ mean the convolution integration over the variables of the momentum fractions (x, z, x 3 ) and the conjugate space coordinates b i of k iT . With the help of the two-pion distribution amplitudes, many works have been done for quasi-two-body decays, the parameters in the S-wave and P -wave two-pion distribution amplitudes have been fixed in Refs. [42,43]. Based these work, we have studied the S-wave resonance contributions to the decays B 0 [45], and the P -wave resonance (ρ(770)) contributions to B 0 (s) → (D/P )ρ → (D/P )ππ decays [46,47] with D represents the charmed D mesons and the P stands for the light pseudoscalar mesons: π, K, η or η ′ .
Up to now, several decay modes of the B and B s mesons to the charmonium state plus pion pair, like B 0 → J/ψπ + π − [1, [16][17][18], B 0 s → J/ψπ + π − [14,15], B 0 (s) → ψ(2S)π + π − [20] and B 0 s → η c π + π − [21], have been measured by BaBar and LHCb Collaboration. With the continuous running of the LHCb experiment, more data of such B/B s decays with the inclusion of various excited charmonium states ( η c (2S) etc.) will be collected. It is therefore interesting to study such decay modes theoretically. In this work, we will study the S-wave resonance contributions to B 0 (s) → η c (2S)f 0 (X) → η c (2S)π + π − decays and give our predictions for the branching fractions of the considered decay modes. This paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. The numerical values, some discussions and the conclusions will be given in last two sections.

II. THE THEORETICAL FRAMEWORK
In the B 0 (s) → η c (2S)π + π − decays, by using of the light-cone coordinates and in the rest frame of B 0 (s) meson, the momentum of B 0 (s) , the pion pair and η c (2S) could be chosen as , r = m ηc(2S) /m B and ω 2 = p 2 means the squared invariant mass of the pion pair. The momenta for the spectators in the B 0 (s) meson, the pion pair, and the η c (2S) meson read as where the momentum fractions x B , z, and x 3 run from zero to unity.
The S-wave two-pion distribution amplitudes can be written as [42,67] with n + = (1, 0, 0 T ), n − = (0, 1, 0 T ) and the π + meson momentum fraction ζ = p + 1 /p + . Their asymptotic forms are parameterized as [42] with the time-like scalar form factor F s (w 2 ) and the Gegenbauer coefficient a I=0 2 = 0.2 ± 0.2. The expressions of the time-like scalar form factor F s (ω 2 ) associated with the ss component of both f 0 (980) and f 0 (1500), and dd component of f 0 (500) can be found in Ref. [42]. Following the LHCb collaboration [14][15][16][17], the Breit−Wigner (BW) formula for the f 0 (500) and f 0 (1500) resonances will be used to parameterize the time-like scalar form factors in the two-pion distribution amplitudes, which include both the resonant and non-resonant contributions of the ππ pair. For f 0 (980), however, the Flatté model [68] will be used since f 0 (980) is close to the KK threshold and the BW formula does not work well for this meson [68,69]. We know that there exist some disputations about the nature of the meson f 0 (500) due to its wide shape. Following the same treatment of f 0 (500) as LHCb collaboration [19], we here also parameterize its contribution to the scalar form factor in the Bugg resonant line-shape [69] with the following relevant parameters In the numerical calculation, we set m r = 0.953 GeV , s A = 0.41 m 2 π , b 1 = 1.302 GeV , b 2 = 0.340 GeV −1 , A = 2.426 GeV 2 and g 4π = 0.011 GeV [69]. The phase-space factors of the decay channels ππ, KK and ηη are defined as ρ i (s) = 1 − 4m 2 i /s with i = 1, 2, 3 for π, K and η respectively. It is worth of mentioning that another description of pion-pion form factors were introduced in Ref. [70,71].
For the B 0 (s) mesons, we use the same distribution amplitudes φ B (x, b) in the b space as being used for example in Ref. [44], The distribution amplitude is chosen as In the numerical calculation, we also use the shape parameter ω B = 0.40 ± 0.04 GeV with f B = 0.19 GeV for B 0 decays, and ω Bs = 0.50 ± 0.05 GeV with f Bs = 0.236 GeV for B 0 s decays [44]. As the first radial excitation of the η c charmonium ground state, η c (2S) is observed firstly by the Belle collaboration in B decays [72,73]. The harmonic-oscillator wave function with the principal quantum number n = 2 and the orbital angular momentum l = 0 is defined as [74] The asymptotic models for the twist-2 distribution amplitudes ψ v , and the twist-3 distribution amplitudes ψ s for the radially excited η c (2S) is parameterized as [75] Ψ , and (S − P )(S + P ) currents, respectively. The total decay amplitudes for the considered decays can therefore be written as where C i (µ)(i = 1, ..., 10) are Wilson coefficients at the renormalization scale µ. For simplicity, we denote the distribution amplitudes Φ I=0 vν=− (z, ζ, ω 2 ) [Φ I=0 s (z, ζ, ω 2 ), Φ I=0 tν=+ (z, ζ, ω 2 )] by φ 0 (φ s , φ σ ) below. From Fig. 1(a) and 1(b), we find with a color factor C F = 4/3. The explicit expressions of the hard functions h a and h b , the evolution factors E e (t i ) including the Sudakov exponents and the hard scales (t a , t b ) can be found for example in Ref. [42]. Following the same procedure, one can obtain the explicit expressions for decay amplitude M LL , M LR and M SP from the evaluation of Fig. 1(c) and 1(d).
From our numerical calculations, we find the following results: • In Fig. 2(a), we show the differential branching ratios dB/dω for B 0 s → η c (2S)π + π − decay, where the solid curve and the dots curve shows the contribution from f 0 (980) and f 0 (1500) is taken into account, respectively. In Fig. 2(b), we show the ω-dependence of the differential decay rate dB/dω when the BW model (solid curve) and the Bugg's model (dots curve) are employed. The allowed region of ω is 4m 2 π ≤ ω 2 ≤ (M B − m ηc(2S) ) 2 .
• For the decays B 0 s → η c (2S)f 0 (X) → η c (2S)π + π − , when the contribution from f 0 (980) and f 0 (1500) are included respectively, the PQCD predictions for the branching ratios where the first two errors come from the uncertainty ω Bs = 0.50 ± 0.05 GeV and a I=0 2 = 0.2 ± 0.2, the last two errors are from w = 0.2 ± 0.1 GeV and f ηc(2S) = 0.243 +0.079 −0.111 GeV ( the parameters in the wave function of η c (2S)). The errors from the uncertainties of other input parameters, for instance the CKM matrix elements, are very small and have been neglected.
By taking into account the S-wave contributions from f 0 (980) and f 0 (1500) simultaneously, we find the PQCD prediction for the total branching ratio: It is easy to see that the dominant contribution comes from the resonance f 0 (980) (82.0%), while the constructive interference between f 0 (980) and f 0 (1500) provide ∼ 13% enhancement to the total decay rate. One can read out this information from Fig. 2(a) approximately. When compared with the previous study for B 0 s → η c (π + π − ) s in Ref. [44], we find that B(B → η c (2S)[π + π − ] s ) : B(B → η c [π + π − ] s ) ≈ 1 : 2.
• For B 0 → η c (2S)f 0 (500) → η c (2S)π + π − decay, the PQCD predictions based on the BW model or the Bugg's model for the parametrization of the wide f 0 (500) are the following: where the major errors have been added in quadrature. One can see easily that the PQCD predictions obtained by employing the BW model or the Bugg's model are very similar, the difference is only about 10%.
• Based on our previous studies of the quasi-two-body B meson decays involving ρ meson [43], we get to know that the main contribution lies indeed in the region around the pole mass of the ρ resonance. Because Γ ηc(2S) ≈ 11.3 MeV is much narrow than Γ ρ ≈ 149 MeV, it is reasonable for us to assume that the possible effect due to the narrow width of η c (2S) is very small and can be neglected safely.